The suppression of random parameter on the boundary crisis of the smooth and discontinuous oscillator system

In this paper, the effect of random parameter on the boundary crisis of the smooth and discontinuous oscillator system is studied. First of all, we convert this stochastic system into a high-dimensional equivalent deterministic system by means of the orthogonal polynomial approximation. Afterward, the method of composite cell coordinate system is used to exhibit the global dynamical behavior of this system in different state spaces. Then, the attractors, basins of attraction and saddles of this system are obtained. We find that there exists interesting boundary crisis phenomenon in this system and the randomness of parameter has an obvious effect on it. Finally, we make a conclusion that the random parameter could suppress the boundary crisis of this system.